Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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520 9 General Processes: Mathematical Facts
The quadratic covariation [M,N] is the limit in probability of
p(n)
i=0
(M
t
n
i
− M
t
n
i−1
)(N
t
n
i
− N
t
n
i−1
) (9.3.4)
when n goes to infinity and sup(t
n
i
− t
n
i−1
) goes to zero, where the sequence
t
n
i
satisfies 0 = t
n
0
<t
n
1
< ···<t
n
p(n)+1
= t.
Let M and N be local martingales equal to 0 at time 0, such that the
product MN is a special semi-martingale (see Subsection 9.3.4). We denote
by M,N (mixed predictable bracket) the unique predictable process
with finite variation such that MN −M,N is a local martingale equal to 0
at time 0. If M is locally square integrable, we denote by M its predictable
bracket, that is M = M,M.
We shall see later under which sufficient condition does the mixed
predictable bracket exist. In particular, if M and N are continuous, the
mixed predictable bracket exists and is the (continuous) quadratic covariation
process defined in Subsection 1.3.1.
If M and N are two local martingales and if the predictable bracket M,N
exists, then the process [M,N] −M,N is a local martingale.
Example 9.3.2.3 If M is the compensated martingale associated with a
Poisson process N and if W is a Brownian motion with respect to the same
filtration
[W, W ]
t
= t, [M,M]
t
= N
t
, [W, M]
t
=0.
Example 9.3.2.4 Let (N
t
,t ≥ 0) be a Poisson process with constant
intensity λ,andM the associated compensated martingale.
(a) Let f and g be two square integrable functions (or predictable square
integrable processes) and
X
t
=
t
0
f(s)dM
s
,Y
t
=
t
0
g(s)dM
s
.
Then,
[X, Y ]
t
=
s≤t
f(s)g(s)ΔN
s
, X, Y
t
=
t
0
f(s)g(s)λds .
(b) Let (Y
i
,i≥ 1,Z
i
,i≥ 1) be i.i.d. random variables, independent of N,
and define the compound Poisson processes
U
t
=
N
t
i=1
Y
i
,V
t
=
N
t
i=1
Z
i
.
Then
[U, V ]
t
=
N
t
i=1
Y
i
Z
i
, U, V
t
= tλE(Y
1
Z
1
) .
9.3 Stochastic Integration for Semi-martingales 521
9.3.3 Orthogonality
Definition 9.3.3.1 Two local martingales are orthogonal if their product is
a local martingale.
In particular, if M and N are independent, they are orthogonal in the
filtration generated by M and N . The converse does not hold (see Exercise
1.5.2.2). We illustrate this fact with a new example.
Let Z = X + iY a complex Brownian motion (see Subsection 5.1.3). Then,
the two processes A
t
=
t
0
(X
s
dY
s
− Y
s
dX
s
)andρ
t
=
t
0
(X
s
dX
s
+ Y
s
dY
s
)=
(|Z
t
|
2
−2t)/2 are orthogonal, but not independent: indeed they have the same
predictable variation process which is not deterministic, as would be the case
if the martingales A and ρ were independent.
Example 9.3.3.2 We present further examples of orthogonal martingales.
• The two local martingales X and Y in Example 9.3.2.4 are orthogonal if
fg =0,ds× P a.s..
• If M
U
and M
V
are the compensated martingales associated with the
compound Poisson process defined in Example 9.3.2.4, they are orthogonal
whenever E(Y
1
Z
1
) = 0. Note that the covariation process [M
U
,M
V
]isnot
equal to 0, it is a local martingale.
• If X is a L´evy process (see Chapter 11) without drift and continuous
part, with L´evy measure ν, the martingales
s≤t
f(ΔX
s
) −t
ν(dx)f(x)
and
s≤t
g(ΔX
s
) − t
ν(dx)g(x)(see Exercise 11.2.3.13) are orthog-
onal iff
ν(dx)f(x)g(x) = 0 and are independent iff fg =0.
Definition 9.3.3.3 A local martingale X is purely discontinuous (a
compensated sum of jumps) if X
0
=0and if it is orthogonal to any continuous
local martingale.
The preceding definition is justified by the following: from Corollary
9.3.7.2, any martingale with bounded variation is orthogonal to any continuous
martingale. We emphasize again, as we did in Warning 9.2.1.3, that the
expression purely discontinuous martingale comes as a whole, and one should
not confuse this notion with that of a purely discontinuous bounded variation
process.
Example 9.3.3.4 If N is a Poisson process of intensity λ, the compensated
martingale (M
t
= N
t
− λt, t ≥ 0) is a purely discontinuous martingale. In
that case,
s≤t
ΔM
s
= N
t
.
Theorem 9.3.3.5 (Canonical Decomposition.) Every local martingale
M can be uniquely decomposed as follows: M = m + M
c
+ M
d
where M
c
is a continuous local martingale, M
c
0
=0, M
d
is a purely discontinuous local
martingale M
d
0
=0and m = M
0
is an F
0
-measurable random variable.
522 9 General Processes: Mathematical Facts
Example 9.3.3.6 The Az´ema martingale μ
t
=(sgnB
t
)
√
t − g
t
(see Propo-
sition 4.3.8.1) is a purely discontinuous martingale in its own filtration. From
Exercise 4.3.8.3, its predictable bracket is t/2. Its continuous martingale part
equals 0 and its quadratic variation process is
[μ]
t
=
s≤t
(Δμ
s
)
2
= g
t
.
It is important to note that
s≤t
|Δμ
s
| = ∞, P a.s.(seeProtter[727]fora
proof). The Az´ema martingale satisfies the equation
d[μ]
t
=
dt
2
− μ
t−
dμ
t
.
Indeed,
μ
2
t
=2
t
0
μ
s−
dμ
s
+[μ]
t
that is t − g
t
=2
t
0
μ
s
−
dμ
s
+ g
t
or g
t
= −
t
0
μ
s−
dμ
s
+
t
2
which leads
immediately to [μ]
t
=
t
2
−
t
0
μ
s−
dμ
s
. This is an example of the so-called
structure equations, which are equations of the form
d[M,M]
t
= α
t
dt + Φ
t
dM
t
,
where M is required to be a local martingale and α and Φ are predictable
functionals of M. The particular case
d[M,M]
t
= dt + βM
t
−
dM
t
,
admits a unique weak solution. For −2 ≤ β<0, [M, M]
c
=0andM is
bounded.
Comment 9.3.3.7 See Subsection 10.6.2, Dellacherie et al. [241], Emery
[325, 326], Protter [727] and Chapter 15 in Yor [868] for a general study of
structure equations.
9.3.4 Semi-martingales
We give several results on semi-martingales. We refer the reader to Meyer
[647]orProtter[727] for the proofs of these results.
Definition 9.3.4.1 An F-semi-martingale is a c`adl`ag process X which can
be written as X = M + A where M is an F-local martingale and where A is
an F-adapted c`adl`ag process with finite variation with value 0 at time 0.
In general, this decomposition is not unique, and we shall speak about
decompositions of semi-martingales. It is necessary to add some conditions
on the finite variation process to get the uniqueness.
9.3 Stochastic Integration for Semi-martingales 523
Definition 9.3.4.2 A special semi-martingale is a semi-martingale with
a predictable finite variation part. Such a decomposition X = M + A with A
predictable, is unique. We call it the canonical decomposition of X,ifit
exists.
The class of semi-martingales is stable under time-changes (see Section 5.1)
and mutual absolute continuity changes of probability measures. Moreover, if
X is a semi-martingale, then f(X) is a semi-martingale if and only if f
is a
Radon measure (see C¸ inlar et al. [189]).
Example 9.3.4.3 Local martingales, super-martingales, finite variation pro-
cesses, c`adl`ag adapted processes with independent and stationary increments,
ItˆoandL´evy processes are semi-martingales.
Examples of non-semi-martingales: If B is a Brownian motion, the
process |B
t
|
α
is not a semi-martingale for 0 <α<1 (the second derivative of
the function f(x)=|x|
α
is not a Radon measure).
The process S
t
=
t
0
φ (B
s
) ds where φ does not belong to L
1
loc
(and if
the integral
t
0
φ (B
s
) ds is defined) has zero quadratic variation but infinite
variation, hence is not an F-semi-martingale.
As an example, let S
t
=
t
0
ds
B
s
= lim
ε→0
da
a
L
a
t
1
{|a|≥ε}
. See Subsec-
tion 6.1.2.
Example 9.3.4.4 The Poisson process N with parameter λ is a special semi-
martingale with canonical decomposition N
t
= M
t
+λt. Note that we can also
write a decomposition as N
t
=0+N
t
, where on the right-hand side, N is
considered as an increasing process. Hence, the semi-martingale N admits at
least two (optional) decompositions. (See Chapter 8.)
If |ΔX|≤C where C is a constant, then the semi-martingale X is
special and its canonical decomposition X = M + A satisfies |ΔA|≤C and
|ΔM|≤2C. In particular, if X is a continuous semi-martingale, it is special
and the processes M and A in its canonical decomposition X = M + A are
continuous.
Note that, if F
t
⊂G
t
for every t, i.e., if F is a subfiltration of G,thenanF-
semi-martingale is a G-semi-martingale if and only if any F-local martingale
is a G-semi-martingale (see Section 5.9). We recall the following result of
Stricker [807]:
Proposition 9.3.4.5 Let F and G be two filtrations such that for all t ≥ 0,
F
t
⊂G
t
.IfX is a G-semi-martingale which is F-adapted, then it is also an
F-semi-martingale.
Let X be a semi-martingale such that ∀t ≥ 0,
s≤t
|ΔX
s
| < ∞. The
process (X
t
−
s≤t
ΔX
s
,t≥ 0) is a continuous semi-martingale with unique
decomposition M + A where M is a continuous local martingale and A a
524 9 General Processes: Mathematical Facts
continuous process with bounded variation. The continuous martingale M
is called the continuous martingale part of X, it is denoted by X
c
,and
[X, X]
c
=[X
c
,X
c
]=X
c
. Note however that not every semi-martingale
satisfies
s≤t
|ΔX
s
| < ∞ and looking for the continuous martingale part of
X may be more complicated (see Chapter 11). If X is a martingale such
that
s≤t
|ΔX
s
| < ∞, it does not imply in general that the processes X
and (X
c
t
+
s≤t
ΔX
s
,t ≥ 0) are equal. In fact, X
t
= X
c
t
+
s≤t
ΔX
s
if
and only if
s≤t
ΔX
s
is a local martingale. This is a strong condition: for
example, under the assumption ΔX
s
= H
s
1
{ΔX
s
=0}
with H predictable, this
condition is satisfied only for continuous martingales. Indeed, in that case, the
process
s≤t
(ΔX
s
)
2
=
s≤t
H
s
ΔX
s
is a positive local martingale, hence a
super-martingale, its expectation equals 0, hence the continuity of X follows.
Proposition 9.3.4.6 If X and Y are semi-martingales and if X
c
,Y
c
are
their continuous martingale parts, their quadratic covariation is
[X, Y ]
t
= X
c
,Y
c
t
+
s≤t
(ΔX
s
)(ΔY
s
) .
Proof: In a first step, we note that [X
c
]=[X]
c
, and we extend this
equality by polarization. Then, we note that the jumps of [X, Y ]
t
and
X
c
,Y
c
t
+
s≤t
ΔX
s
ΔY
s
are the same.
9.3.5 Stochastic Integration for Semi-martingales
If X is a semi-martingale with decomposition M +A, then for any predictable,
locally bounded process H, we can define the process
(HX)
t
:=
t
0
H
s
dX
s
=
t
0
H
s
dM
s
+
t
0
H
s
dA
s
,
where
t
0
H
s
dA
s
is a Stieltjes integral.
The process (HX)
t
does not depend on the decomposition of the semi-
martingale X. It is a semi-martingale, in particular it is a c`adl`ag adapted
process. The map H → HX is linear and the map X → HX is linear, the
process HX is a semi-martingale.
If X is a semi-martingale and H a predictable process, the jump process
of the stochastic integral of H with respect to X is equal to H times the
jump process of X:(Δ(HX))
t
= H
t
(ΔX)
t
. It can also be checked that
HX
c
=(HX)
c
.
Comment 9.3.5.1 See Dellacherie and Meyer [244] Chapter 8, Jacod [468],
Jacod and Shiryaev [471] Chapter 1, Kallenberg [505] and Protter [727]
Chapter 2, for more information on stochastic integration.
9.3 Stochastic Integration for Semi-martingales 525
9.3.6 Quadratic Covariation of Two Semi-martingales
We examine which of the previous constructions and properties of [X, Y ]may
be extended to pairs of semi-martingales. The quadratic covariation of two
semi-martingales is the finite variation process defined by the integration by
parts formula
[X, Y ]=XY − X
0
Y
0
− X
−
Y − Y
−
X .
For every predictable bounded process H, the quadratic covariation of
Y and of HX (the stochastic integral of H with respect to X), is the
stochastic integral of H with respect to the quadratic covariation of X and
Y :[HX,Y ]=H[X, Y ].
If either X or Y has locally finite variation, then the sum
0<s≤t
|ΔX
s
||ΔY
s
|
is almost surely finite and the quadratic covariation of the pair (X, Y ) is given
by [X, Y ]
t
= X
0
Y
0
+
0<s≤t
ΔX
s
ΔY
s
. For general semi-martingales X and
Y ,wehave
0<s≤t
|ΔX
s
||ΔY
s
|≤
⎛
⎝
0<s≤t
(ΔX
s
)
2
⎞
⎠
1/2
⎛
⎝
0<s≤t
(ΔY
s
)
2
⎞
⎠
1/2
hence the sum
0<s≤t
|ΔX
s
||ΔY
s
| is almost surely finite.
Note that, if P and Q are equivalent, the quadratic covariations of the
semi-martingales X and Y under P and under Q are the same.
An adapted increasing process A is said to be a compensator for the
semi-martingale Y if Y − A is a local martingale. For example if X is a
local martingale, the process [X, X] is a compensator for X
2
. In general,
a semi-martingale admits many compensators. If there exists a predictable
compensator, then it is unique (among predictable compensators). Once more,
we note that in general, [X, X] is optional and not predictable.
Remark 9.3.6.1 If M and N are two pure jump martingales with the same
jumps, they are equal. Indeed, the difference M −N is a continuous martingale
which is orthogonal to itself, hence is equal to 0.
9.3.7 Particular Cases
The integration by parts formula
X
t
Y
t
= X
0
Y
0
+
t
0
X
s−
dY
s
+
t
0
Y
s−
dX
s
+[X, Y ]
t
,
can be simplified, as presented in Yoeurp [857], under some additional
hypotheses.
526 9 General Processes: Mathematical Facts
Proposition 9.3.7.1 Let X be a semi-martingale.
(a) If A is a bounded variation process
X
t
A
t
= X
0
A
0
+
t
0
X
s
dA
s
+
t
0
A
s−
dX
s
(9.3.5)
and [X, A]=ΔXA.
(b) If A is a predictable process with bounded variation
X
t
A
t
= X
0
A
0
+
t
0
X
s−
dA
s
+
t
0
A
s
dX
s
(9.3.6)
and [X, A]=ΔAX.
Proof: Part a: If A is a bounded variation process
t
0
X
s−
dA
s
+[X, A]
t
=
t
0
X
s−
dA
s
+
s≤t
ΔX
s
ΔA
s
=
t
0
(X
s−
+ ΔX
s
)dA
s
=
t
0
X
s
dA
s
.
Part b: If M is a martingale and A a predictable process with bounded
variation, Yoeurp [857] established that
[M,A]
t
=
s≤t
ΔM
s
ΔA
s
=
n,T
n
≤t
ΔA
T
n
ΔM
T
n
=
t
0
n
ΔA
T
n
1
{T
n
=s}
dM
s
=
t
0
ΔA
s
dM
s
where (T
n
) is a sequence of stopping times which exhaust the jumps, therefore,
the process [M, A]
t
is a local martingale. If A is a predictable process with
bounded variation
t
0
A
s−
dX
s
+[X, A]
t
=
t
0
A
s−
dX
s
+
s≤t
ΔX
s
ΔA
s
=
t
0
A
s−
dX
s
+
t
0
(A
s
− A
s−
)dX
s
=
t
0
A
s
dX
s
.
We recall that the notation X
mart
= Y means that X − Y is a local
martingale.
9.3 Stochastic Integration for Semi-martingales 527
Corollary 9.3.7.2 Assume that M is a local martingale and A abounded
variation process. Then
M
t
A
t
mart
=
t
0
M
s
dA
s
.
Consequently, if M is continuous and A is a local martingale with bounded
variation, the process MA is also a local martingale.
If A is a predictable bounded variation process,
M
t
A
t
mart
=
t
0
M
s−
dA
s
.
Then, here, we recover partly the orthogonality between continuous local
martingales and purely discontinuous martingales (Definition 9.3.3.3).
9.3.8 Predictable Bracket of Two Semi-martingales
We first discuss the predictable bracket of two martingales, which we have
defined earlier. We now give some conditions for the existence of this bracket.
If the quadratic covariation process is locally integrable, then the predictable
bracket exists. In particular this is the case if X and Y are locally square
integrable and X, Y may be defined as the dual predictable projection of
[X, Y ].
If X is a semi-martingale such that X = M + A where M is locally square
integrable and A predictable, one can define its predictable bracket as follows.
Note that
[M + A]
t
=[M]
t
+2[M, A]
t
+[A]
t
.
We have seen in Proposition 9.3.7.1 that the process [M, A]
t
=
t
0
ΔA
s
dM
s
is
a local martingale, hence [M +A]
t
mart
= M
t
+[A]
t
. Since [A] is predictable, the
process X := M +[A] is predictable too and is called the predictable
bracket of the semi-martingale X.IfX is a continuous semi-martingale,
X
t
= M
t
.
In the general case, one defines X
t
as the dual predictable projection of
[X]
t
. The polarization formula is used to define the mixed predictable bracket
X, Y .
Warning 9.3.8.1 In general, if X = M + A is a semi-martingale with
predictable bracket X, the process X
2
t
−X
t
is not a martingale. Indeed,
X
2
t
−X
t
= M
2
t
+ A
2
t
+2M
t
A
t
−M
t
− [A]
t
.
528 9 General Processes: Mathematical Facts
Using
A
2
t
+2M
t
A
t
− [A]
t
=2M
t
A
t
+2
t
0
A
s−
dA
s
mart
=2
t
0
X
s−
dA
s
one obtains
X
2
t
mart
= X
t
+2
t
0
X
s−
dA
s
.
If P and Q are equivalent probabilities, the mixed predictable bracket
of the continuous semi-martingales X and Y is the same under these
two probabilities. This is no longer true if discontinuous martingales are
considered. (See the case of Poisson processes in Comment 8.2.4.2.)
9.4 Itˆo’s Formula and Girsanov’s Theorem
We give here, without proof, a general Itˆo formula (see Protter [727]fora
proof).
9.4.1 Itˆo’s Formula: Optional and Predictable Forms
Itˆo’s formula is an extremely powerful tool. It is, therefore, worthwhile writing
it in different ways. We recall that we have discussed Itˆo’s formula for
continuous semi-martingales in Subsection 1.5.3.
If X is a semi-martingale, X
c
its continuous martingale part and f a C
2
(R)
function, then the sum
s≤t
|f(X
s
)−f(X
s
−
)−f
(X
s
−
)ΔX
s
| is almost surely
finite for any t and one has the optional Itˆoformula
f(X
t
)= f(X
0
)+
t
0
f
(X
s
−
)dX
s
+
1
2
t
0
f
(X
s
)dX
c
s
+
0<s≤t
[f(X
s
) − f (X
s
−
) − f
(X
s
−
)ΔX
s
] .
Note that in the particular case where X is continuous, we recover Itˆo’s
formula established in Theorem 1.5.3.1:
f(X
t
)=f(X
0
)+
t
0
f
(X
s
)dX
s
+
1
2
t
0
f
(X
s
)dX
s
.
Coming back to the general case, we recall that
X
c
t
=[X
c
]
t
=[X]
c
t
=[X]
t
−
s≤t
(ΔX
s
)
2
.
9.4 Itˆo’s Formula and Girsanov’s Theorem 529
If X is a special semi-martingale, and [X] is locally integrable, then, for any
f ∈ C
2
, f(X
t
) is a special semi-martingale.
More generally, let X =(X
1
,...,X
d
) be a semi-martingale and f a C
2
(R
d
)
function. Let us denote by M
i,c
the continuous martingale part of X
i
. Then,
f(X
t
)=f(X
0
)+
d
i=1
t
0
∂
i
f(X
s
−
)dX
i
s
+
1
2
t
0
d
i,j=1
∂
ij
f(X
s
−
) dM
i,c
,M
j,c
s
+
s≤t
f(X
s
) − f (X
s
−
) −
d
i=1
∂
i
f(X
s
−
)ΔX
i
s
. (9.4.1)
Obviously, this formula entails the form of Itˆo’s formula for f(M
t
,A
t
)where
M is a multidimensional martingale, and A a multidimensional process with
bounded variation.
Definition 9.4.1.1 AprocessX is a pseudo-continuous semi-martin-
gale (PCSM) if its additive decomposition is X = M + V where M is a local
martingale and V is an adapted process with finite variation such that:
(i) V is continuous (hence, predictable),
(ii) ΔX
s
= H
s
1
{ΔX
s
=0}
where H is predictable,
(iii) the predictable compensator A of
s≤t
(ΔX
s
)
2
=
s≤t
(ΔM
s
)
2
=
s≤t
H
2
s
1
{ΔX
s
=0}
is continuous.
In particular, PCSM’s are special semi-martingales. Note that, if X is a
PCSM, the predictable compensator of
s≤t
Φ(ΔX
s
)1
{ΔX
s
=0}
=
s≤t
Ψ(H
s
)(ΔX
s
)
2
is
t
0
Ψ(H
s
)dA
s
,whereΨ(x)=
Φ(x)
x
2
1
{x=0}
.
If X is a PCSM with decomposition X
t
= M
c
t
+ M
d
t
+ V
t
,andf asmooth
function, then f(X) is a PCSM. We present the canonical decomposition of
the special semi-martingale f(X): from Itˆo’s formula
f(X
t
)=f(X
0
)+
t
0
f
(X
s
−
)dX
s
+
1
2
t
0
f
(X
s
−
)dM
c
,M
c
s
+
s≤t
[f(X
s
−
+ H
s
) − f (X
s
−
) − f
(X
s
−
)H
s
] 1
{ΔX
s
=0}
,
= f (X
0
)+
t
0
f
(X
s
−
)dX
s
+
1
2
t
0
f
(X
s
−
)dM
c
,M
c
s
+
s≤t
f
(T )
(X
s
−
,H
s
)(ΔX
s
)
2
,